Properties

Label 507.2.e.e
Level $507$
Weight $2$
Character orbit 507.e
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{7} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{7} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{11} - q^{12} + 6 q^{14} + 5 \zeta_{12}^{2} q^{16} + ( -6 + 6 \zeta_{12}^{2} ) q^{17} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{21} + ( -6 + 6 \zeta_{12}^{2} ) q^{22} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{24} -5 q^{25} - q^{27} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -6 \zeta_{12}^{2} q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{32} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{33} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{34} -\zeta_{12}^{2} q^{36} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{37} + 6 q^{38} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{41} + 6 \zeta_{12}^{2} q^{42} + ( -4 + 4 \zeta_{12}^{2} ) q^{43} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{44} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} + ( -5 + 5 \zeta_{12}^{2} ) q^{48} -5 \zeta_{12}^{2} q^{49} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{50} -6 q^{51} + 6 q^{53} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{54} + ( 6 - 6 \zeta_{12}^{2} ) q^{56} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{58} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{59} + ( 2 - 2 \zeta_{12}^{2} ) q^{61} + 6 \zeta_{12}^{2} q^{62} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} + q^{64} -6 q^{66} + ( 6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12}^{2} q^{68} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{71} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{72} + ( 12 - 12 \zeta_{12}^{2} ) q^{74} -5 \zeta_{12}^{2} q^{75} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{76} + 12 q^{77} -8 q^{79} -\zeta_{12}^{2} q^{81} + ( 12 - 12 \zeta_{12}^{2} ) q^{82} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{84} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{86} + ( 6 - 6 \zeta_{12}^{2} ) q^{87} + 6 \zeta_{12}^{2} q^{88} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{93} + 6 \zeta_{12}^{2} q^{94} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{96} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{97} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{98} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{4} - 2q^{9} - 4q^{12} + 24q^{14} + 10q^{16} - 12q^{17} - 12q^{22} - 20q^{25} - 4q^{27} - 12q^{29} - 2q^{36} + 24q^{38} + 12q^{42} - 8q^{43} - 10q^{48} - 10q^{49} - 24q^{51} + 24q^{53} + 12q^{56} + 4q^{61} + 12q^{62} + 4q^{64} - 24q^{66} - 12q^{68} + 24q^{74} - 10q^{75} + 48q^{77} - 32q^{79} - 2q^{81} + 24q^{82} + 12q^{87} + 12q^{88} + 12q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 0.500000 0.866025i −0.500000 0.866025i 0 0.866025 + 1.50000i −1.73205 3.00000i −1.73205 −0.500000 0.866025i 0
22.2 0.866025 1.50000i 0.500000 0.866025i −0.500000 0.866025i 0 −0.866025 1.50000i 1.73205 + 3.00000i 1.73205 −0.500000 0.866025i 0
484.1 −0.866025 1.50000i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.866025 1.50000i −1.73205 + 3.00000i −1.73205 −0.500000 + 0.866025i 0
484.2 0.866025 + 1.50000i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.866025 + 1.50000i 1.73205 3.00000i 1.73205 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.e 4
13.b even 2 1 inner 507.2.e.e 4
13.c even 3 1 507.2.a.f 2
13.c even 3 1 inner 507.2.e.e 4
13.d odd 4 1 507.2.j.a 2
13.d odd 4 1 507.2.j.c 2
13.e even 6 1 507.2.a.f 2
13.e even 6 1 inner 507.2.e.e 4
13.f odd 12 2 39.2.b.a 2
13.f odd 12 1 507.2.j.a 2
13.f odd 12 1 507.2.j.c 2
39.h odd 6 1 1521.2.a.l 2
39.i odd 6 1 1521.2.a.l 2
39.k even 12 2 117.2.b.a 2
52.i odd 6 1 8112.2.a.bv 2
52.j odd 6 1 8112.2.a.bv 2
52.l even 12 2 624.2.c.e 2
65.o even 12 2 975.2.h.f 4
65.s odd 12 2 975.2.b.d 2
65.t even 12 2 975.2.h.f 4
91.bc even 12 2 1911.2.c.d 2
104.u even 12 2 2496.2.c.d 2
104.x odd 12 2 2496.2.c.k 2
156.v odd 12 2 1872.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.f odd 12 2
117.2.b.a 2 39.k even 12 2
507.2.a.f 2 13.c even 3 1
507.2.a.f 2 13.e even 6 1
507.2.e.e 4 1.a even 1 1 trivial
507.2.e.e 4 13.b even 2 1 inner
507.2.e.e 4 13.c even 3 1 inner
507.2.e.e 4 13.e even 6 1 inner
507.2.j.a 2 13.d odd 4 1
507.2.j.a 2 13.f odd 12 1
507.2.j.c 2 13.d odd 4 1
507.2.j.c 2 13.f odd 12 1
624.2.c.e 2 52.l even 12 2
975.2.b.d 2 65.s odd 12 2
975.2.h.f 4 65.o even 12 2
975.2.h.f 4 65.t even 12 2
1521.2.a.l 2 39.h odd 6 1
1521.2.a.l 2 39.i odd 6 1
1872.2.c.e 2 156.v odd 12 2
1911.2.c.d 2 91.bc even 12 2
2496.2.c.d 2 104.u even 12 2
2496.2.c.k 2 104.x odd 12 2
8112.2.a.bv 2 52.i odd 6 1
8112.2.a.bv 2 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 9 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 144 + 12 T^{2} + T^{4} \)
$11$ \( 144 + 12 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 36 + 6 T + T^{2} )^{2} \)
$19$ \( 144 + 12 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 36 + 6 T + T^{2} )^{2} \)
$31$ \( ( -12 + T^{2} )^{2} \)
$37$ \( 2304 + 48 T^{2} + T^{4} \)
$41$ \( 2304 + 48 T^{2} + T^{4} \)
$43$ \( ( 16 + 4 T + T^{2} )^{2} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( -6 + T )^{4} \)
$59$ \( 11664 + 108 T^{2} + T^{4} \)
$61$ \( ( 4 - 2 T + T^{2} )^{2} \)
$67$ \( 11664 + 108 T^{2} + T^{4} \)
$71$ \( 144 + 12 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( ( -12 + T^{2} )^{2} \)
$89$ \( 2304 + 48 T^{2} + T^{4} \)
$97$ \( 36864 + 192 T^{2} + T^{4} \)
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